Additive Lie ($ξ$-Lie) Derivations and Generalized Lie ($ξ$-Lie) Derivations on Prime Algebras

The additive (generalized) $ξ$-Lie derivations on prime algebras are characterized. It is shown, under some suitable assumption, that an additive map $L$ is an additive (generalized) Lie derivation if and only if it is the sum of an additive (generalized) derivation and an additive map from the algebra into its center vanishing all commutators; is an additive (generalized) $ξ$-Lie derivation with $ξ\not=1$ if and only if it is an additive (generalized) derivation satisfying $L(ξA)=ξL(A)$ for all $A$. These results are then used to characterize additive (generalized) $ξ$-Lie derivations on several operator algebras such as Banach space standard operator algebras and von Neumman algebras.

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